Rényi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields

Abstract

One-parameter functionals of the Rényi $R_{\rho ,\gamma }(\alpha )$ and Tsallis $T_{\rho ,\gamma }(\alpha )$ types are calculated both in the position (subscript $\rho$) and momentum ($\gamma$) spaces for the azimuthally symmetric 2D nanoring that is placed into the combination of the transverse uniform magnetic field $\mathbf{B}$ and the Aharonov-Bohm (AB) flux ${\varphi }_{AB}$ and whose potential profile is modelled by the superposition of the quadratic and inverse quadratic dependencies on the radius $r$. Position (momentum) Rényi entropy depends on the field $B$ as a negative (positive) logarithm of ${\omega }_{eff}\equiv {\left({\omega }_0^2+{\omega }_c^2/4\right)}^{1/2}$, where ${\omega }_0$ determines the quadratic steepness of the confining potential and ${\omega }_c$ is a cyclotron frequency. This makes the sum ${R_{\rho }}_{nm}(\alpha )+{R_{\gamma }}_{nm}(\frac{\alpha }{2\alpha -1})$ a field-independent quantity that increases with the principal $n$ and azimuthal $m$ quantum numbers and does satisfy corresponding uncertainty relation. Analytic expression for the lower boundary of the semi-infinite range of the dimensionless coefficient $\alpha$ where the momentum entropies exist reveals that it depends on the ring geometry, AB intensity and quantum number $m$. It is proved that there is the only orbital for which both Rényi and Tsallis uncertainty relations turn into the identity at $\alpha =1/2$ and which is not necessarily the lowest-energy level. At any coefficient $\alpha$, the dependence of the position Rényi entropy on the AB flux mimics the energy variation with ${\varphi }_{AB}$ what, under appropriate scaling, can be used for the unique determination of the associated persistent current. Similarities and differences between the two entropies and their uncertainty relations are discussed too.

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• Entropic measures of Rydberg-like harmonic states
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Cited by in corpus (7)

• Quantum-information theory of magnetic field influence on circular dots with different boundary conditions
• Thermodynamic properties of a charged particle in non-uniform magnetic field
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• One-dimensional pseudoharmonic oscillator: classical remarks and quantum-information theory
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots
• Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field
• Information theoretic measures on quantum droplets in ultracold atomic systems